Scaled first–order methods for a class of large–scale constrained least square problems

Coli, Vanna Lisa; Ruggiero, Valeria; Zanni, Luca

doi:10.1063/1.4965314

Cited by 4 publications

(4 citation statements)

References 17 publications

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“…Proposition 3.1 ensures that SGP algorithm converges without restrictive assumptions on the step-length parameter α k and the diagonal scaling matrix D k , whose choices can be directed to accelerate the convergence rate of the scheme. Even if the theoretical convergence rate O(1/k) on the objective function values is lower than the rate O(1/k 2 ) of some optimal first-order methods exploiting extrapolation/inertial steps [28,5,2,21], the practical performance of SGP method, achievable by suitable selections of D k and α k , is very well comparable with the convergence rate of the optimal algorithms [9,30,8,13]. In the following we provide the updating rules for D k and α k that allow SGP to efficiently solve problem (5).…”

Section: A Scaled Gradient Approach For Ct Image Reconstructionmentioning

confidence: 98%

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Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm

et al. 2017

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We propose a scaled gradient projection algorithm for the reconstruction of 3D X-ray tomographic images from limited data. The problem arises from the discretization of an ill-posed integral problem and, due to the incompleteness of the data, has infinite possible solutions. Hence, by following a regularization approach, we formulate the reconstruction problem as the nonnegatively constrained minimization of an objective function given by the sum of a fit-to-data term and a smoothed differentiable Total Variation function. The problem is challenging for its very large size and because a good reconstruction is required in a very short time. For these reasons, we propose to use a gradient projection method, accelerated by exploiting a scaling strategy for defining gradient-based descent directions and generalized Barzilaiâ\u80\u93Borwein rules for the choice of the step-lengths. The numerical results on a 3D phantom are very promising since they show the ability of the scaling strategy to accelerate the convergence in the first iterations

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Section: A Scaled Gradient Approach For Ct Image Reconstructionmentioning

confidence: 98%

“…Following the suggestions in [8,13], the parameter ρ k+1 is chosen as ρ k+1 = 1 + 10 15 /(k + 1) 2.1 .…”

Section: A Scaled Gradient Approach For Ct Image Reconstructionmentioning

confidence: 99%

Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm

et al. 2017

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“…Downloaded 05/03/20 to 129.175.97.14. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php It has been shown in [13] and [8] that the presence of a variable metric in firstorder methods can significantly improve the performance in solving problem (5.10) with respect to their standard nonscaled versions. For this reason we only report the results obtained by comparing the variable metric GP method SGP with different choices for the step length parameter.…”

Section: Reconstruction Of Fiber Orientation Distribution In Diffusio...mentioning

confidence: 99%

Spectral Properties of Barzilai--Borwein Rules in Solving Singly Linearly Constrained Optimization Problems Subject to Lower and Upper Bounds

Crisci¹,

Porta²,

Ruggiero³

et al. 2020

SIAM J. Optim.

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In 1988, Barzilai and Borwein published a pioneering paper which opened the way to inexpensively accelerate first-order. In more detail, in the framework of unconstrained optimization, Barzilai and Borwein developed two strategies to select the step length in gradient descent methods with the aim of encoding some second-order information of the problem without computing and/or employing the Hessian matrix of the objective function. Starting from these ideas, several efficient step length techniques have been suggested in the last decades in order to make gradient descent methods more and also more appealing for problems which handle large-scale data and require realtime solutions. Typically, these new step length selection rules have been tuned in the quadratic unconstrained framework for sweeping the spectrum of the Hessian matrix, and then applied also to nonquadratic constrained problems, without any substantial modification, by showing them to be very effective anyway. In this paper, we deeply analyze how, in quadratic and nonquadratic minimization problems, the presence of a feasible region, expressed by a single linear equality constraint together with lower and upper bounds, influences the spectral properties of the original Barzilai-Borwein (BB) rules, generalizing recent results provided for box-constrained quadratic problems. This analysis gives rise to modified BB approaches able not only to capture second-order information but also to exploit the nature of the feasible region. We show the benefits gained by the new step length rules on a set of test problems arising also from machine learning and image processing applications.

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“…In this work, we propose to solve (1) by an accelerated gradient scheme belonging to the class of Scaled Gradient Projection (SGP) methods 18,19 . The SGP methods have been recently applied in low-sampled X-rays cone beam CT (CBCT) image reconstruction, with very good results in terms of image accuracy 13,14,20 . In particular, in 14 the authors proposed a SGP method for X-rays CT image reconstruction and applied it to a phantom simulation using a geometry different from DBT limited angles.…”

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confidence: 99%

GPU acceleration of a model-based iterative method for Digital Breast Tomosynthesis

Cavicchioli

Piccolomini

et al. 2020

Sci Rep

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Digital Breast Tomosynthesis (DBT) is a modern 3D Computed Tomography X-ray technique for the early detection of breast tumors, which is receiving growing interest in the medical and scientific community. Since DBT performs incomplete sampling of data, the image reconstruction approaches based on iterative methods are preferable to the classical analytic techniques, such as the Filtered Back Projection algorithm, providing fewer artifacts. In this work, we consider a Model-Based Iterative Reconstruction (MBIR) method well suited to describe the DBT data acquisition process and to include prior information on the reconstructed image. We propose a gradient-based solver named Scaled Gradient Projection (SGP) for the solution of the constrained optimization problem arising in the considered MBIR method. Even if the SGP algorithm exhibits fast convergence, the time required on a serial computer for the reconstruction of a real DBT data set is too long for the clinical needs. In this paper we propose a parallel SGP version designed to perform the most expensive computations of each iteration on Graphics Processing Unit (GPU). We apply the proposed parallel approach on three different GPU boards, with computational performance comparable with that of the boards usually installed in commercial DBT systems. The numerical results show that the proposed GPU-based MBIR method provides accurate reconstructions in a time suitable for clinical trials. Digital Breast Tomosynthesis (DBT) is a quite recent 3D Computed Tomography (CT) technique providing a volumetric breast reconstruction as a stack of 2D images, each representing a cross-sectional slice of the breast itself 1. When compared with traditional 2D mammography, DBT has the advantage of separating the breast anatomical tissues, which can be overlapped in a mammography, and this generally reduces false negative diagnosis. At the same time, since the DBT source emits X-rays only from a small number of angles in an arc trajectory, DBT provides a low radiation dose comparable to the radiation dose used in a standard mammography. For this reason, DBT is appealing to the medical and scientific community as a breast screening routine 2,3. Until a few years ago, the introduction of DBT in a clinical setting has been slowed down by the absence of efficient algorithms for image reconstruction from a limited number of projections. It is well known that the traditional Filtered Back Projection (FBP) 4 analytic algorithm amplifies noise and artifacts in the case of low-sampled data 3. Iterative algorithms are known to perform better than FBP in the case of tomosynthesis data, but they require computational times incompatible with a clinical use 3,5 , since in clinical DBT examinations the breast reconstruction must be provided in 40 to 60 seconds. Nevertheless, the recent advent of low cost parallel architectures, such as Graphics Processing Units (GPUs), has provided the chance for a remarkable reduction of the image reconstruction time, making iterative methods a realistic alterna...

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Scaled first–order methods for a class of large–scale constrained least square problems

Cited by 4 publications

References 17 publications

Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm

Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm

Spectral Properties of Barzilai--Borwein Rules in Solving Singly Linearly Constrained Optimization Problems Subject to Lower and Upper Bounds

GPU acceleration of a model-based iterative method for Digital Breast Tomosynthesis

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